Probability and the Full House

One of the prettiest hands is the full house. As far as poker terminology goes, it has reached farthest into common parlance. Getting dealt a full house is considered a lucky occurance, but it is frequent enough to be within the realm of reasonable hope. Big families, big cards, big bets. So what are the chances of being dealt a full house? First we need to compute how many combinations of the 52 cards will result in a full house at the average poker table, using the average poker deck with no wild cards. The following equation will give us this number:

(13 X 4) X (12 X 6) = 3744

The 4 in our equation is the number of combinations possible in each card type (A, K, Q, etc.) that would bring about 3 of a kind. (The four combinations for the 3, for example would be this: 3 of spades, 3 of hearts, 3 of diamonds together, then - 3 of spades, 3 of diamonds, 3 of clubs, then - 3 of spades, 3 of hearts, 3 of clubs, finally - 3 of hearts, 3 of diamonds, 3 of clubs). The 13 represents the number of suits. The six represents the combinations possible in pairs (two cards chosen from four possible choices) and the 12 is the card types remaining after one is used up for the three of a kind part of the hand. Now, we take these possible hands and divide it into the total possible of all hands:

3744/2,598,960 = 1 in 694

This, of course, is the possibility you have of actually being dealt a full house. It is unlikely but not as remote as the previous hands we have examined. Here again there are many opportunities for moving up from a lesser hand. Most notably relying on two pair to turn into a full house. The chances are pretty good, for there are going to be 47 unknown cards to choose from and 4 of these will make the house full. Thus, your chances are 4/47 at this point. Almost worth raising on - better than cutting the deck and hoping for an ace.